5.1 The chain rule for functions of several variables

The chain rule for functions of several variables builds on what you already know about derivatives. It's like leveling up your calculus skills, allowing you to handle more complex situations involving multiple variables.

This rule is super useful when dealing with composite functions. It helps you break down tricky problems into manageable pieces, making it easier to find derivatives and understand how different variables affect each other.

Derivatives of Composite Functions

The Chain Rule and Composite Functions

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• The chain rule extends the concept of the single-variable chain rule to functions of several variables
  • Enables finding the derivative of a composite function by multiplying the partial derivatives of the outer function with the gradient of the inner function
• A composite function is a function that depends on another function
  • For example, if $f(x, y) = x^2 + y^2$ and $g(t) = (t, t^3)$, then $f(g(t)) = t^2 + t^6$ is a composite function
• To apply the chain rule, express the composite function as a composition of an outer function and an inner function
  • The outer function takes the output of the inner function as its input
  • The inner function is a function of the original independent variables

Partial Derivatives and the Total Derivative

• Partial derivatives measure the rate of change of a function with respect to one variable while holding other variables constant
  • For a function $f(x, y)$, the partial derivative with respect to $x$ is denoted as $\frac{\partial f}{\partial x}$ and the partial derivative with respect to $y$ is denoted as $\frac{\partial f}{\partial y}$
• The total derivative is the generalization of the single-variable derivative to functions of several variables
  • It measures the rate of change of a function in any direction
  • For a function $f(x, y)$, the total derivative is given by $df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$
• When applying the chain rule, the total derivative of the composite function is the product of the partial derivatives of the outer function and the total derivatives of the inner functions
  • For example, if $f(x, y) = x^2 + y^2$ and $x = g(t) = t$ and $y = h(t) = t^3$, then $\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt} = 2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2t(1) + 2t^3(3t^2) = 2t + 6t^5$

Multivariable Calculus Fundamentals

Multivariable Functions and Differentiability

• Multivariable functions are functions that depend on multiple variables
  • For example, $f(x, y) = x^2 + y^2$ is a function of two variables, $x$ and $y$
• A multivariable function is differentiable at a point if its partial derivatives exist and are continuous at that point
  • Differentiability ensures that the function is smooth and well-behaved near the point
• Differentiability allows for the approximation of a function using a linear function (its total derivative) near a point
  • This linear approximation is the basis for many applications, such as optimization and error estimation

Gradient Vector and Jacobian Matrix

• The gradient vector of a multivariable function $f(x_1, x_2, \ldots, x_n)$ is a vector of its partial derivatives
  • It points in the direction of the greatest rate of increase of the function
  • The gradient vector is denoted as $\nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \ldots, \frac{\partial f}{\partial x_n}\right)$
• The Jacobian matrix is a matrix of partial derivatives for a vector-valued function $\mathbf{f}(x_1, x_2, \ldots, x_n) = (f_1(x_1, x_2, \ldots, x_n), f_2(x_1, x_2, \ldots, x_n), \ldots, f_m(x_1, x_2, \ldots, x_n))$
  • It generalizes the gradient vector to vector-valued functions
  • The Jacobian matrix is denoted as $J_{\mathbf{f}} = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}$
• The gradient vector and Jacobian matrix play crucial roles in optimization, coordinate transformations, and analyzing the behavior of multivariable functions